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Annual Conference of ITAAnnual Conference of ITAACITA 2009ACITA 2009
Neighbor Discovery in Wireless Networks and the Coupon Collector’s Problem
Sudarshan Vasudevan (UMass), Don Towsley (UMass), Dennis Goeckel (UMass) and Ramin Khalili (EPFL)*
Problem Definition
• Nodes have just been placed/thrown/dropped• Every node has a unique ID• Nodes are beginning to power up• How does each node determine the IDs of its
neighbors to begin ad hoc network formation?• Challenge: Nothing is known at the time of deployment
What is missing?
• No apriori transmission scheduling among nodes can cause interference
• No prior knowledge of node density• Lack of synchronization among nodes• Detecting when to start and terminate neighbor
discovery phase non-trivial
Idle: No discovery Collision: No discovery One-and-only-one transmit: That ID is discovered (by all)
Typical Approach
• ALOHA – each node transmits with probability p
Prior Work
• Aloha-like ND Algorithms [MMcGlynn01,SVasudevan05,SBorbash07] • Focus on determining optimal p
• Time required to discover all neighbors under optimal settings unknown
• Assume prior information about node density• ND initiation/termination not handled
Aloha-like Discovery
• Assume node density known and perfect synchronization among nodes
• Nodes cannot distinguish between collision and idle time slot (hence, a node does not know when it has been discovered)
• Each node transmits with p = 1/n• Our Key Observation: Reduces to Coupon
Collector’s Problem• Time to discover all neighbors )ln ( nn
Removing Assumptions
• Unknown n: Algorithm execution in phases• In phase j, transmit with probability • Only a factor 2 slowdown from knowing n
• Asynchronism: Collision duration doubled• Each phase 2 times longer• Factor of 2 slowdown from synchronous execution
• Account for clock skews to allow different start times• Termination condition based on number of nodes
discovered in each phase
j21
Collision Detection-based ND
• Each node sends feedback of reception status• Once a node has been discovered by its neighbors, it
stops transmitting• A factor of ln n improvement in time to discover
neighbors, which is • Order optimal
• Unknown node density, asynchronous execution and initiation/termination• Handled similar to Aloha-like neighbor discovery
Conclusions
• Result: a ND algorithm running in time when there is no feedback and time when nodes provide feedback of reception status, with no assumptions on:• Knowledge of number of neighbors• Synchronization among nodes• Initial starting time• Knowledge of when to terminate
)(n
)ln ( nn)(n
* Portion of work done when the author was a post-doctoral researcher at UMass Amherst